European Journal of Mechanics B/Fluids 29 (2010) 442–450

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European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Effect of streamwise and spanwise electric fields on transient growth in atwo-fluid channel flow

Fang Li ∗, Xie-Yuan Yin, Xie-Zhen YinDepartment of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, People’s Republic of China

a r t i c l e i n f o

Article history:Received 28 May 2009Received in revised form8 June 2010Accepted 10 June 2010Available online 19 June 2010

Keywords:Electric fieldTransient growthChannel flowInstability

a b s t r a c t

A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both thefluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwiseand spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. Thenumerical result shows that the streamwise electric field suppresses transient growth of the disturbancewith spanwise uniform wave number. The spanwise electric field diminishes transient growth of thedisturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in thewave number plane. The peak at relatively large spanwise wave number is dominated by the lift-upmechanism and little influenced by electric field. Differently, the peak at relatively small wave numberis associated with the characteristic of the interface and possibly influenced by electric field. The effectof the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth isstudied aswell. A scaling law is obtained for relatively smallWeber numbers and relatively large Reynoldsnumbers.

© 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Transient behavior of channel flow has attracted much atten-tion for the past decades [1–4]. It is found that small disturbancesmay be amplified significantly at initial time and trigger tran-sition to turbulence. Mathematically, transient growth is due tothe non-normality of linear operator and the non-orthogonality ofeigenfunctions. Physically, it is due to the lift-up mechanism. Sev-eral strategies have been put forward for the purpose of enhanc-ing/diminishing transient growth in channel flow, for instance,Airiau and Castets [5] and Krasnov et al. [6] imposed a magneticfield to suppress energy growth, Biau and Bottaro [7] and Sameenand Govindarajan [8] evaluated thermal controlling in increasingor decreasing themagnitude of transient growth, and Li et al. [9] in-vestigated the effect of a normal electric field on transient behaviorof a two-fluid channel flow. In the current paperwe extendour pre-vious work to the cases of streamwise and spanwise electric fields.The target is to evaluate and compare the effect of different electricfields on transient growth in two-fluid channel flow. The paper isorganized as follows: in Section 2 the model is described and theequations are given; in Section 3 the numerical result is shown andthe effect of the streamwise and spanwise electric fields on tran-sient growth is studied; in Section 4 the main conclusion is drawn.

∗ Corresponding author.E-mail address: fli6@ustc.edu.cn (F. Li).

0997-7546/$ – see front matter© 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2010.06.003

2. Model and formulation

2.1. Model and basic state

Our theoretical model is based on that proposed by Yecko [10].As sketched in Fig. 1, the channel flow consists of two incompress-ible viscous fluids of densitiesρ1, ρ2 and viscositiesµ1, µ2, respec-tively. The depths of the lower and upper fluid layers areH1 andH2,respectively. The effect of gravity is neglected. The interface ten-sion between the fluids is denoted by γ . Both the fluids are perfectdielectrics with dielectric constants ε1 and ε2, respectively. Thereis no free charge in bulk or at interface. The flow is driven by aconstant pressure gradient in x direction. Taking H1 and U0 (thevelocity at the interface) as the scales of length and velocity, re-spectively, the nondimensional basic velocity profile is [10]

U1(y) = −µr + HrHr (Hr + 1)

y2 +H2r − µrHr (Hr + 1)

y+ 1,

− 1 ≤ y ≤ 0, (1)

U2(y) = −µr + Hr

µrHr (Hr + 1)y2 +

H2r − µrµrHr (Hr + 1)

y+ 1,

0 ≤ y ≤ Hr , (2)

where µr = µ2/µ1 and Hr = H2/H1. The flow is subjected to auniform electric field either in x direction (the streamwise electricfield case) or in z direction (the spanwise electric field case). Themagnitude of electric field is denoted by E1 in the streamwise

F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450 443

Fig. 1. Sketch of the two-fluid channel flow.

electric field case and by E2 in the spanwise electric field case. Dueto the absence of free charge, electric field has no influence on thebasic velocity profile. Moreover, electric field influences the linearstability of the systemonly through the normal force balance at theinterface [9].

2.2. Solutions to electric field perturbations

Introduce an electric potential perturbation φj satisfying theLaplace equation ∇2φj = 0, where the subscript j = 1, 2 denotesthe lower and upper fluids respectively and ∇2 = ∂2/∂x2 +∂2/∂y2 + ∂2/∂z2. The perturbation of electric field intensity Ej =−∇φj [11].The system is perturbed by an infinitesimal three-dimensional

disturbance of the form ϕ (x, y, z, t) = ϕ̂ (y, t) ei(αx+βz), whereϕ represents the perturbation of any physical quantity and ϕ̂ isthe corresponding eigenfunction, α and β are the wave numbersin streamwise and spanwise directions respectively, and i isthe imaginary unit. Hence the eigenfunction of electric potentialperturbation φ̂j satisfies the following partial differential equation

∂2φ̂j

∂y2− k2φ̂j = 0,

where k2 = α2 + β2. The solutions are

φ̂1 = C1 sinh (ky)+ C2 cosh (ky) , −1 ≤ y ≤ 0,

φ̂2 = C3 sinh (ky)+ C4 cosh (ky) , 0 ≤ y ≤ Hr ,

where C1 − C4 are the coefficients to be determined by boundaryconditions with respect to electric field. The boundary conditionsinclude the disappearance of the electric potential perturbationat the upper and lower walls, the continuity of the tangentialcomponent of electric field intensity at the interface, and thecontinuity of the normal component of electric displacement at theinterface [9]. After straightforward calculation, we obtain

C1 = iδk−1θ (εr − 1) f̂ , C2 = iδk−1θ tanh(k) (εr − 1) f̂ ,

C3 = −iδk−1θ tanh(k) tanh−1 (Hrk) (εr − 1) f̂ ,

C4 = iδk−1θ tanh(k) (εr − 1) f̂ ,

where δ is equal to α for the streamwise electric field case andto β for the spanwise electric field case, sinh, cosh and tanh arethe hyperbolic functions, θ = (1+ εr tanh(k)/ tanh (Hrk))−1, therelative electrical permittivity εr = ε2/ε1, and f̂ is the amplitudeof the interface perturbation f

(= f̂ (t) ei(αx+βz)

).

2.3. Linearized governing equations and boundary conditions

The linearized governing equations of the flow can be expressedin terms of the normal velocity perturbation v̂j and the normalvorticity perturbation η̂j, i.e.[(

∂

∂t+ iαUj

) (D2 − k2

)− iαU ′′j −

1Rej

(D2 − k2

)2]v̂j = 0, (3)

[(∂

∂t+ iαUj

)−1Rej

(D2 − k2

)]η̂j = −iβU ′j v̂j, (4)

where D2 = ∂2/∂y2,U ′ = dU/dy,U ′′ = d2U/dy2, the Reynoldsnumbers Re1 = Re = ρ1U0H1/µ1 and Re2 = Reρr/µr (ρr = ρ2/ρ1).The boundary conditions at the upper and lower walls are

v̂j = Dv̂j = η̂j = 0, (5)where D = ∂/∂y. At the interface the kinematic condition shouldbe satisfied, i.e.(ddt+ iαUj

)f̂ = v̂j; (6)

the velocity is continuous in normal, streamwise and spanwisedirections, i.e. [10]

v̂2 = v̂1, (7)

α(Dv̂2 − Dv̂1

)− β

(η̂2 − η̂1

)− ik2

(U ′2 − U

′

1

)f̂ = 0, (8)

β(Dv̂2 − Dv̂1

)+ α

(η̂2 − η̂1

)= 0; (9)

and the force is balanced in two tangential and one normal direc-tions, i.e.

µr

[α(D2 + k2

)v̂2 − βDη̂2 − ik2U ′′2 f̂

]−

[α(D2 + k2

)v̂1 − βDη̂1 − ik2U ′′1 f̂

]= 0, (10)

µr[β(D2 + k2

)v̂2 + αDη̂2

]−[β(D2 + k2

)v̂1 + αDη̂1

]= 0, (11)

ρr

[(∂

∂t+ iαU2

)Dv̂2 − iαU ′2v̂2

]−

[(∂

∂t+ iαU1

)Dv̂1 − iαU ′1v̂1

]−µr(D3v̂2 − 3k2Dv̂2

)Re

+D3v̂1 − 3k2Dv̂1

Re

−

(k4

We+ kδθ tanh(k) (εr − 1)2

)f̂ = 0, (12)

where the Weber number We = ρ1U20H1/γ , the electrical Eulernumbers ζ = ε1E21/ρ1U

20 and ξ = ε1E22/ρ1U

20 , and δ is equal to

α2ζ for the streamwise electric field case and to β2ξ for the span-wise electric field case.The governing equations (3)–(4) and the boundary condi-

tions (5)–(12) constitute an initial-value problem. The initial-valueproblem is finally transformed into a generalized eigenvalue prob-lem. The corresponding eigenvalues and eigenfunctions are ob-tained by means of Chebyshev spectral collocation method. AMatlab code is developed to solve the problem. The validity of thecode has been checked.

2.4. Energy norm

A proper energy norm should be chosen to evaluate transientgrowth. The choice of energy norm is somewhat delicate, whichis discussed in the next section. Here we choose an energy normincluding kinetic energy, electric energy and interfacial potentialenergy due to surface tension. The norm is expressed as

‖q‖2E =12k2

[∫ Hr

0ρr

(∣∣Dv̂2∣∣2 + k2 ∣∣v̂2∣∣2 + ∣∣η̂2∣∣2) dy+

∫ 0

−1

(∣∣Dv̂1∣∣2 + k2 ∣∣v̂1∣∣2 + ∣∣η̂1∣∣2) dy]+12k2

We|f̂ |2 +

12δθk−1 tanh(k) (εr − 1)2 |f̂ |2, (13)

444 F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450

Fig. 2. Effect of the streamwise electric field on (a) the eigenvalue spectrum and(b) the transient growth function. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100and εr = 2.

where δ is equal toα2ζ for the streamwise electric field case and toβ2ξ for the spanwise electric field case. Note that electric energyand interfacial potential energy are expressed as a function of theinterface displacement. It is derived from a volume integral. (Fordetails see Ref. [9].) The transient growth function is defined as

G(t) = supq(0)6=0

‖q(t)‖2E‖q(0)‖2E

.

The optimal growth in time is GO = supt≥0 G(t), and thepeak value of GO in wave number plane is GP = GO (αP , βP) =supα,β GO (α, β), where αP and βP are the corresponding wavenumbers.

3. Result and discussion

In this section the effect of the streamwise and spanwise electricfields on the transient growth of the two-fluid channel flow isstudied. The result is compared with that of the normal electricfield case. The effect of the Weber number, the Reynolds numberand the relative electrical permittivity on transient growth is alsostudied.

3.1. Effect of streamwise and spanwise electric fields on transientgrowth

It can be seen from Eqs. (12) and (13) that the streamwiseelectric field has no influence on the disturbance with streamwise

Fig. 3. Effect of the spanwise electric field on (a) the eigenvalue spectrum and (b)the transient growth function. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100 andεr = 2.

uniform wave number (i.e. α = 0) and that the spanwise electricfield has no influence on the disturbance with spanwise uniformwave number (i.e. β = 0). Therefore, in the calculation a two-dimensional disturbance (α = 1, β = 0) is selected for thestreamwise electric field case and a three-dimensional disturbancewith streamwise uniformwave number (α = 0, β = 1) is selectedfor the spanwise electric field case. The eigenvalue spectrum incomplex ω plane is shown in Fig. 2(a) for the streamwise electricfield case and in Fig. 3(a) for the spanwise electric field case. InFig. 2(a), the least stable mode is the interfacial one having thelargest phase velocity. The streamwise electric field stabilizes itsgrowth rate and increases its phase velocity. The second leaststable mode is the interfacial one having the smallest phasevelocity. The streamwise electric field destabilizes the mode whiledecreasing its phase velocity. The effect of the streamwise electricfield on the interfacial modes is contrary to that of the normalelectric field. (See Fig. 2 in [9].) In Fig. 3(a), the spanwise electricfield has no evident influence on the modes except two interfacialones having the same absolute value of phase velocity. Differentfrom the normal electric field, the spanwise electric field stabilizesthe interfacial modes and increases the absolute value of theirphase velocities.The evolution of the transient growth function G(t) in time is

shown in Fig. 2(b) for the streamwise electric field case and inFig. 3(b) for the spanwise electric field case. In Fig. 2(b), the optimalgrowth GO is decreased by the streamwise electric field, although

F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450 445

Fig. 4. Contours of the optimal growth GO in (α, β) plane. (a) The streamwiseelectric field case, ζ = 2, ξ = 0; (b) the spanwise electric field case, ζ = 0, ξ = 0.1.ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100 and εr = 2.

its initial growth rate is slightly increased. The asymptotic growthat large time is decreased by the electric field. Considering thatasymptotic growth ismainly associatedwith the least stablemode,the interfacial modes sensitive to electric field are indicated to bethe least stable. In Fig. 3(b), the optimal growth is decreased by thespanwise electric field. However, the asymptotic behavior at largetime is little influenced by the electric field, indicating that theleast stablemode is not interfacial. Apparently, the streamwise andspanwise electric fields influence the transient growth in two-fluidchannel flow in a way different from the normal electric field. Thenormal electric field was found to enhance the transient growthof both two-dimensional and three-dimensional disturbances. (SeeFig. 3 in [9].)Comparing Fig. 2(b) with Fig. 3(b), we can see that a three-

dimensional disturbance with streamwise uniform wave numberhas optimal growth much larger than a two-dimensional distur-bance. In order to better understand transient behavior of differ-ent disturbances, the contours of GO in (α, β) plane are drawn inFig. 4(a) for the streamwise electric field case and in Fig. 4(b) for thespanwise electric field case. The range of α and β is from 0 to 5. Thegray area in the figures indicates the asymptotic unstable region inwhich the least stable mode has positive growth rate. ComparingFig. 4 with Fig. 4(a) in [9] where electric field is absent, the asymp-totic unstable region is shrunk by the streamwise electric field but

Fig. 5. Optimal disturbance at (a) t = 0 and (b) t = tO in (y, z) plane, u contours,(v,w) vectors, and the interface displacement f (the solid line near y = 0),corresponding to peak B in Fig. 4(b).

little influenced by the spanwise electric field. There are two peaksin the wave number plane. Both of them possess streamwise uni-form wave number regardless of the direction and magnitude ofelectric field. One peak is GP = 72.2 at βP = 3.2, denoted by sym-bol A in the figure. The other is GP = 59.4 at βP = 1.0 in Fig. 4(a)and GP = 35.2 at βP = 0.77 in Fig. 4(b), denoted by symbol B. PeakA is hardly influenced by electric field since it is dominated by thelift-up mechanism, as outlined in [9]. The streamwise electric fieldhas negligible influence on peak B. The spanwise electric field di-minishes the peak value andwave number of peak B. Nevertheless,the spanwise electric field influences peak B in awaydifferent fromthe normal electric field [9]. The configuration of optimal distur-bance at peak B is plotted in Fig. 5. Clearly, the optimal disturbanceat initial time is two corotating streamwise vortices, while at opti-mal time it turns into a single row of vortices with the center justbelow the interface. Moreover, the interface displacement is sup-pressed, indicating that interfacial potential energy is transformedinto kinetic energy. The configuration of optimal disturbance is ba-sically consistent with that shown in Fig. 7 in [9] where electricfield is absent.The effect of the spanwise electric field on peak B is further in-

vestigated in Fig. 6. Fig. 6(a) and (b) illustrate the wave numberof optimal disturbance βP and the peak value of optimal growthGP respectively as the function of the electrical Euler number ξ .Apparently, both βP and GP decrease monotonically with increas-ing the spanwise electric field intensity. When the electric field issufficiently strong, both βP and GP approach certain asymptotic

446 F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450

Fig. 6. (a) The spanwise wave number βP and (b) the energy growth GP at peakB versus the electrical Euler number ξ in the spanwise electric field case. Thevalues of the Weber number are 1000, 400, 100, 40 and 10 from the top down.ρr = 0.9, µr = 2,Hr = 1, Re = 900 and εr = 2.

Fig. 7. Contours of the optimal growth GO in (α, β) plane in the absence of electricfield. ρr = 0.9, µr = 2,Hr = 1, Re = 900 andWe = 1000.

Fig. 8. Optimal growth at peak A versus the Reynolds number in the absence ofelectric field. ρr = 0.9, µr = 2 and Hr = 1.

values. The effect of the spanwise electric field is similar for dif-ferent values of the Weber number.In Fig. 6(b) the peak value at peak B exceeds that at peak A

when theWeber number is large (We ≥ 400 approximately). Fromthis point the characteristic of the peaks at large Weber numbersneeds to be further investigated. Fig. 7 shows the contours of GO in(α, β) plane atWe = 1000. Compared with Fig. 4, the asymptoticunstable region in the figure is enlarged, indicating that theWebernumber has a destabilization effect on asymptotic instability. Onthe other hand, only one peak exists (GP = 103.1 at βP = 1.75,denoted by symbol C). The calculation of the configuration ofoptimal disturbance shows that peak C is identical to peak B. Thatis, peak A disappears at large Weber numbers.

3.2. Effect of Weber number and Reynolds number on transientgrowth

Two-fluid channel flow is more complicated than single-fluidchannel flow. In single-fluid channel flow there is usually oneparameter, i.e. the Reynolds number. In two-fluid channel flowmore parameters can be involved. For example, in this problemthree parameters related to the upper and lower fluid layers, i.e.ρr , µr and Hr , and one parameter related to the interface, i.e. theWeber number, are involved. Therefore it is muchmore difficult tostudy the scaling law in two-fluid channel flow.In two-fluid channel flow there are twopeaks of optimal growth

in wave number plane: one is dominated by lift-up mechanismand the other is associatedwith interface characteristic. Yecko [10]concluded that the former is independent of the Weber numberand the latter is reduced by surface tension according to a scalinglaw ofW 1/2e while 1 < We < 50. In the following we recheck thescaling law involving the Weber number and Reynolds number intwo-fluid system. For the sake of convenience, the electric field isset to zero and the other parameters are fixed as before.Since theWeber number has no effect on peak A, we only check

the Reynolds number. As shown in Fig. 8, the optimal growth atpeak A has a linear relationship with the Reynolds number inlogarithmic plot. The slope of the curve is about 2, that is, GP ∝ R2e .In their study of transient behavior of two-phase mixing layer,Yecko and Zaleski [12] also found that the optimal energy growthscales with the square of the Reynolds number, as in single-phaseflow. On the other hand, the wave number of optimal disturbanceβP at peak A is little influenced by the Reynolds number.

F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450 447

Fig. 9. (a) The spanwise wave number βP and (b) the energy growth GP at peak Bversus the Weber number in the absence of electric field. ρr = 0.9, µr = 2 andHr = 1.

Fig. 10. Optimal growth at peak B versus the Reynolds number in the absence ofelectric field. ρr = 0.9, µr = 2 and Hr = 1.

The effect of the Weber number on βP and GP of peak B isillustrated in Fig. 9 for different values of the Reynolds number.In Fig. 9(a) the wave number of optimal disturbance βP increases

Fig. 11. Effect of the relative electrical permittivity on (a) the eigenvalue spectrumand (b) the transient growth function in the streamwise electric field case. ρr =0.9, µr = 2,Hr = 1, Re = 900,We = 100 and ζ = 2.

monotonously with the Weber number. It can also be seen thatβP decreases with the Reynolds number increasing. In Fig. 9(b) thepeak value of optimal growth GP is influenced significantly by boththe Weber number and the Reynolds number. When the Webernumber is relatively small, GP has a linear relationship with theWeber number, i.e. GP ∝ W ne . The value of n increases with theReynolds number. In the range of Reynolds number investigated, nis between 0.3 and 0.5.Fig. 10 illustrates the optimal growth at peak B as the function of

the Reynolds number. As illustrated in the figure, the scaling lawwith respect to the Reynolds number is influenced by the Webernumber.When theWeber number is relatively small (We < 40, therange is enlarged as the Reynolds number increases), the optimalgrowth at peak B is proportional to the Reynolds number. The slopeis about 1.0, i.e. GP ∝ Re. As the Weber number increases, thecurves are not linear any more. According to the result in Figs. 9(b)and 10, a scaling law can be formulated as GP ∝ W 0.5e Re forWe <100 and Re > 1000. The scaling lawmay be influenced by the otherparameters, which is out of the scope of this paper.

3.3. Effect of relative electrical permittivity on transient growth

In this model the relative electrical permittivity εr is the onlyparameter related to the electrical property of fluid. It can be seenfrom Eqs. (12) and (13) that when εr is equal to unity (i.e. the lower

448 F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450

Fig. 12. Effect of the relative electrical permittivity on (a) the eigenvalue spectrumand (b) the transient growth function in the spanwise electric field case. ρr =0.9, µr = 2,Hr = 1, Re = 900,We = 100 and ξ = 0.1.

and upper fluids have the same electrical permittivity) neitherthe streamwise nor the spanwise electric field has influence ontransient growth. The same conclusionwas obtained in the normalelectric field case [9]. Figs. 11 and 12 show the effect of εr onthe eigenvalue spectrum and transient growth in the streamwiseand spanwise electric field cases, respectively. For both the casesεr influences the least stable interfacial modes significantly. As itmoves away from unity, the growth rates of the interfacial modesare suppressed and the phase velocities of them are increased.Moreover, in both the streamwise and spanwise cases the transientgrowth is diminished as εr moves away from unity. The effect ofthe relative electrical permittivity on the peak value of optimalgrowth and the corresponding wave number of peak B is shownin Fig. 13(a) and (b), respectively, for the spanwise electric fieldcase. Apparently, as εr moves away from unity, the wave numberof optimal disturbance at peak B is decreased, and so is the peakvalue of optimal growth.

3.4. Discussion on choice of energy norm

The definition of energy norm in initial-value problems becomecomplicated while more than one types of energy are involved.In two-fluid channel flow having a displaced interface, there ex-ist kinetic energy and interfacial potential energy at least. It wasfound that transient growth cannot converge to a finite valuewith-out considering the displacement of interface [13–15]. To solve

Fig. 13. Effect of the relative electrical permittivity on (a) the spanwise wavenumber βP and (b) the energy growth GP of peak B in the spanwise electric fieldcase. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100 and ξ = 0.1.

Fig. 14. Transient growth under different energy norms in the streamwise electricfield case (α = 1, β = 0). Solid: kinetic energy, interface potential energy andelectric energy are all taken into account; dashed: kinetic energy and interfacepotential energy are taken into account; dotted: only kinetic energy is taken intoaccount. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100, εr = 2 and ζ = 2.

F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450 449

Fig. 15. Effect of the streamwise electric field on transient growth (α = 1, β = 0).(a) Both kinetic energy and interface potential energy are taken into account; (b)only kinetic energy is taken into account. ρr = 0.9, µr = 2,Hr = 1, Re =900,We = 100 and εr = 2.

Fig. 16. Transient growth under different energy norms in the spanwise electricfield case (α = 0, β = 1). Solid: kinetic energy, interface potential energy andelectric energy are all taken into account; dashed: kinetic energy and interfacepotential energy are taken into account; dotted: only kinetic energy is taken intoaccount. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We = 100, εr = 2 and ξ = 0.1.

Fig. 17. Effect of the spanwise electric field on transient growth (α = 0, β = 1). (a)Both kinetic energy and interface potential energy are taken into account; (b) onlykinetic energy is taken into account. ρr = 0.9, µr = 2,Hr = 1, Re = 900,We =100 and εr = 2.

the problem of convergence, researchers have proposed three ap-proaches: Malik and Hooper [16] used a miscible layer of variableviscosity to replace the sharp interface between fluids; South andHooper [15] took into account the interface displacement throughdefining a h-norm, and Malik and Hooper [17] modified it into aM-norm; the third approach is to consider directly the potentialenergy of disturbed interface [12,14,18,19]. Schmid [20] pointedout that a physically motivated disturbance measure is the bestchoice despite some shortcomings. Most importantly, the effect ofcapillary on transient growth can be studied through the Webernumber.Nevertheless, it is of interest to study all possible definitions

of energy norm. In this problem there are two other definitions:one is to take into account kinetic energy and interfacial potentialenergy, and the other is to only take kinetic energy into account.Fig. 14 illustrates the result under three energy norms for thestreamwise electric field case and Fig. 16 illustrates the result forthe spanwise electric field case. Apparently, different energy normmight lead to different characteristic of transient growth. Note thatoscillation appears in the figures. De Luca et al. [19] also foundslow and oscillatory decay of transient growth in time. Yecko [10]declared that this oscillatory behavior is due to the participation ofinterfacial modes in non-modal growth.The effect of the streamwise and spanwise electric fields on

transient growth under different energy norms is shown in Figs. 15

450 F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450

and 17, respectively. Comparing Fig. 15 with Fig. 2(b) and Fig. 17with Fig. 3(b), the effect of the streamwise and spanwise electricfields on transient growth is quite different under three energynorms. Particularly, the magnitude of transient growth based onkinematic energy is the largest and the oscillation of it is the mostintense. Further study shows that the period of oscillation is relatedto the least stable eigenvalue in spectrum. On the other hand, it isevident that electric field plays a crucial role in such oscillation ofenergy.

4. Conclusions

The effects of the streamwise and spanwise electric fields ontransient growth of small disturbances in a two-fluid channelflow are investigated. The streamwise electric field is found tosuppress transient growth of two-dimensional disturbances, andthe spanwise electric field diminishes transient growth of three-dimensional disturbances. At relatively small Weber numbers,there are two peaks of optimal growth inwave number plane, bothhaving streamwise uniform wave number; the one at relativelylarge spanwise wave number (peak A) is dominated by the lift-up mechanism and the other at relatively small spanwise wavenumber (peak B) is influenced by the characteristic of interface.The characteristic of peak A is little influenced by the Webernumber or electric field. The peak value of optimal growth andthe corresponding wave number at peak B can be decreased bythe spanwise electric field. At relatively small Weber numbers theoptimal growth at peak A is larger than that at peak B. As theWeber number increases, the optimal growth at peak B is greatlyenhanced and peak Bmay be predominant over peak A. The scalinglaw involving the Weber number and Reynolds number is studiedin the absence of electric field. The optimal growth at peak A obeysa scaling law of the square of the Reynolds number as in single-liquid flow, while the optimal growth at peak B is scaled as GP ∝W 0.5e Re for We < 100 and Re > 1000. The effect of the relativeelectrical permittivity on transient growth is investigated. Thecalculation shows that transient growth decreases as the relativeelectrical permittivity moves away from unity. In addition, thechoice of energy norm is discussed. It is found that different energynorms may lead to different transient growth. Moreover, electricfield may influence the result profoundly.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Project No. 10802084) and the FundamentalResearch Funds for the Central Universities (USTC, Project No.WK2090050006).

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